MinT - Best Known (257−51, 257, s)-Nets in Base 4

Best Known (257−51, 257, s)-Nets in Base 4

(257−51, 257, 1539)-Net over F₄ — Constructive and digital

Digital (206, 257, 1539)-net over F₄, using

t-expansion [i] based on digital (200, 257, 1539)-net over F₄, using
- 1 times m-reduction [i] based on digital (200, 258, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(257−51, 257, 8099)-Net over F₄ — Digital

Digital (206, 257, 8099)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−51, 257, 4961318)-Net in Base 4 — Upper bound on s

There is no (206, 257, 4961319)-net in base 4, because

1 times m-reduction [i] would yield (206, 256, 4961319)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13407 845138 963950 185050 921211 794196 390081 675260 511690 212009 694967 035177 942719 551472 897906 933573 140274 921783 863333 232072 790135 129641 202238 768517 400818 772760 > 4²⁵⁶ [i]